reserve X for RealNormSpace;

theorem Th15:
  for X be RealNormSpace, V be Subset of X, Vt be Subset of
  TopSpaceNorm X st V = Vt holds V is closed iff Vt is closed
proof
  let X be RealNormSpace, V be Subset of X, Vt be Subset of TopSpaceNorm X;
  assume
A1: V = Vt;
  hereby
    assume
A2: V is closed;
    now
      let St be sequence of TopSpaceNorm X;
      assume that
A3:   St is convergent and
A4:   rng St c= Vt;
      reconsider S = St as sequence of X;
      S is convergent by A3,Th13;
      then lim S in V by A1,A2,A4,NFCONT_1:def 3;
      then {lim S} c= V by ZFMISC_1:31;
      hence Lim St c= Vt by A1,A3,Th14;
    end;
    hence Vt is closed by FRECHET:def 7;
  end;
  assume
A5: Vt is closed;
  now
    let S be sequence of X;
    assume that
A6: rng S c= V and
A7: S is convergent;
    reconsider St = S as sequence of TopSpaceNorm X;
A8: St is convergent by A7,Th13;
    then Lim St c= Vt by A1,A5,A6,FRECHET:def 7;
    then {lim S} c= V by A1,A8,Th14;
    hence lim S in V by ZFMISC_1:31;
  end;
  hence thesis by NFCONT_1:def 3;
end;
